Another Hilbert Space Problem

From: Nicolas Ojeda Baer (nojb_at_fibertel.com.ar)
Date: 06/10/04 

Date: 10 Jun 2004 11:11:12 -0700

Greetings.

Let (x_n) be a complete ortogonal system in a hilbert space H (i.e. an
orthogonal basis, not necesarily of unit norm). Now let (y_n) be
another orthogonal sequence in H such that

\sum_{n=1}^\infty ||x_n - y_n|| < 1

Then (y_n) is also complete.

I think that I can prove this in the case in which each x_n is of norm
1 (i.e. orthonormal), but the general case escapes me... Maybe there
is a general proof?

Thanks,
nojb.